# Need help with 'Tsiolkovsky's Rocket Equation': What's the maximum weight of my rockets command module?

A civilization wants to transport a command module towards another planet that is passing by. Their planet has a radius of 6'800 kilometers and weighs about 9 × 10^24 kg. They have no stronger engines than the RD-170. They have an unlimited budget. As the main engines kick in to start fighting against gravity, the rocket is already flying horizontally at about 700 m/s through an atmosphere with about 0.15 bar pressure, having a similar composition to our stratosphere. (I don't know at what height this might be, take 70k feet, if you like) This speed will be reached no matter how heavy the rocket is. Whilst accelerating, it will readjust its wings and drop parts of them to stay on course. Go from there:

What's the maximum weight of my rockets command module?

PS: If there is no maximum to its weight, please give me an equation that describes the rocket-to-payload weight ratio under the circumstances I described.

Edit: Filled in some of the gaps.

Edit2: Removed as much of the narrative as possible

My english isn't the best. Feel free to point out mistakes.

• I'm voting to close this question as off-topic because It describes a pure;y hypothetical situation, of no obvious relevance to actual space travel. It might fit better on the worldbuilding stack exchange – Steve Linton Nov 27 '19 at 16:13
• Your answer depends on a lot of other things (for example, density of the athmosphere). What is "1.5 the material"? Do you think on a planet with 1.5 mass of the Earth? But what would be its radius? – peterh - Reinstate Monica Nov 27 '19 at 17:00
• I'm voting to close this question because it's the second that @justthisonequestion has asked ;) – Punintended Nov 27 '19 at 18:42
• If this is off-topic, we'd have to close 75% of the questions on this site. Hypothetical or not, this is a request for help understanding the Tsiolkovsky equation, which is firmly on-topic in my opinion. – Hobbes Nov 28 '19 at 9:58
• I think the question can be reopened, but a little rewording may be helpful. It took me quite some time to understand the question and go though the comments. – William R. Ebenezer Nov 28 '19 at 12:03

## 1 Answer

Ok. The question is closer to answerable now. I am going to start by making some simplifying assumptions.

1. We'll ignore the atmosphere
2. We'll just consider the problem of getting into a low orbit around the original planet.

So the orbital velocity is $$v_{o}\approx {\sqrt {{\frac {GM}{r}}}}$$ putting in the given values we get about $$9.4 km/s$$. Subtracting the 0.7 km/s you said you started with (you haven't mentioned the planet's rotation, but that might contribute a bit as well) you need $$8.7 km/s$$.

The RD-170 has an $$I_{sp}$$ in vacuum of $$330s$$ so using the rocket equation we get:

$${\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}}}$$ which wa can solve to find $$m_0/m_f = 14.7$$ so to do this with a single stage, your rocket would need to be about 93% propellant at the start, which is just a bit higher than we know how to do at the moment.

A two stage rocket can do this pretty easily. Suppose the first stage adds $$2.24 km/s$$ then the propellant for that must have been half of the initial mass (same equation). Say another 10% for the dry mass of the first stage and you have 40% of your original launch mass for the fuelled second stage. That needs to be about 87% propellant, but it has a much lower thrust requirement, so that is easy.

So a two stage to orbit rocket ignoring atmosphere ends up with about 5% (13% x 40%) payload delivered to orbit, including the empty second stage. That's not amazing, but a 5000 ton rocket, would give you 250 tons on orbit, which is not bad.

The atmosphere does hurt you, both in air resistance, although if you are starting at 0.15 bar that's not the end of the world, and because you need to fight gravity to get above the atmosphere before you go too fast, and because you have to lift your periastron above the atmosphere. On the other hand, the stronger gravity will make the atmospheric density drop off faster with altitude.

These figures are hard to calculate (at least for me) but I would guess you can be at least 1% of launch mass as payload in orbit, and maybe quite a lot more. You could also go to three stages.

Of course orbit is not your original brief. Without knowing much more about this "passing planet" it's hard to say much more. It is possible for instance that you could use a low thrust propulsion system such as a nuclear powered ion drive to lift your orbit slowly until you were "in the way of" the passing planet and then use its atmosphere to brake, in which case a few tons of xenon will go a long way.

• This already helps a lot. thanks. – justthisonequestion Nov 28 '19 at 16:16